Solutions of a nonlinear Schrödinger equation
type: Item
instance of: scholarly article
title: Solutions of a nonlinear Schrödinger equation
zbMATH Open document ID: 1004.35107
author: R. Smith
published in: Discrete and Continuous Dynamical Systems
publication date: 2002-12-02T00:00:00Z
review text: The authors of this interesting paper study the existence and multiplicity of solutions in \(H^1(\mathbb R^N)\) of nonlinear Schrödinger equations, \(\triangle u+V_{\lambda }(x)u=g(x,u)\) in \(\mathbb R^N\). It is assumed that \(u(x)\to 0\) as \(|x|\to \infty \), the potential \(V_{\lambda }(x)=\lambda a(x)+a_0(x)\), where \(\lambda >0\) is a real parameter, \(a_0\in L^{\infty }(\mathbb R^N)\), \(a\in L_{\text{loc}}^{\infty }(\mathbb R ^N)\), and there is a closed subset \(Z\subset \mathbb R^N\) with nonempty \(\Omega =\) int \(Z\) such that \(a(x)=0\) in \(Z\) and \(a(x)>0\) a.e. in \(\mathbb R ^N\setminus Z\), and there is \(M_0>0\) such that the Lebesgue measure mes\((\{x\in \mathbb R ^N:a(x)\leq M_0\})<\infty \). The authors improve some previous results in two respects. The first is that they do not require \(a_0\) to be positive. The second is that \(f(x,u)\) is a critical nonlinear function.
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 1751462
zbMATH Keywords: Schrödinger equation
zbMATH Keywords: multiple solutions
zbMATH Keywords: variational method
reviewed by: Dimitar A. Kolev
MaRDI profile type: Publication
full work available at URL: dcds.2002.8.563
OpenAlex ID: W4230469669
DOI: 10.3934/DCDS.2002.8.563
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description: scientific article; zbMATH DE number 1751462